A formal proof of instability for five-dimensional Schwarzschild black strings

Schwarzschild black holes provide a solution to the four-dimensional Einstein field equations, the mathematical framework that governs general relativity. In higher dimensions, these same equations admit many different solutions. One of these, the five-dimensional Schwarzschild black string, interests theorists because of its apparent instability.

There is numerical evidence that black strings have an unstable mode that grows exponentially when perturbed. In this paper, Sam Collingbourne demonstrates a direct mathematical proof of this instability. By converting the problem to a simpler form and using a variational approach, he shows that the lowest energy solution must produce an unstable mode.

“If you study what’s known as the linearized Einstein vacuum equation around the solution, you can find solutions which grow exponentially in time, which indicates that the solution should be unstable in the more complex nonlinear theory,” said Collingbourne.

Under a carefully chosen perturbation ansatz and suitable gauges, he reduces the linear Einstein vacuum equation to an ordinary differential equation for one function. “Making a gauge choice is key to simplifying the problem,” said Collingbourne. “There were two gauge choices that I [made] in this paper. One is called the transverse traceless, or harmonic gauge condition. The other I call the spherical gauge.” With this last gauge choice, the problem “reduces under an ansatz to a simple Sturm–Liouville eigenvalue problem.”

A change of variables and rescaling takes the resulting ordinary differential equation into a Schrodinger eigenvalue problem. Using a variational approach, Collingbourne shows that the lowest eigenfunction must give rise to an exponentially growing mode.

Collingbourne hopes to use a similar approach to study the Kerr black string, a rotating version of the one studied here.

Source: “The Gregory-Laflamme instability of the Schwarzschild black string exterior,” by Sam C. Collingbourne, Journal of Mathematical Physics (2021). The article can be accessed at https://doi.org/10.1063/5.0043059 .